Question

$\text{If}△ABC\text{is a right triangled with}\angle B={90}^{\circ },\text{and BD is perpendicular to AC,}\text{then AC}\times \text{DC will be equal to:}$

• A. $BC\times AB$
• B. ${BC}^2$
• C. ${BD}^2$
• D. $AB\times AC$

Answer 1

The correct option is B ${BC}^2$

$\text{Given:}\angle ABC={90}^{\circ }\text{BD is perpendicular to AC}\Rightarrow \angle BDC={90}^{\circ }$

$\text{In}△ABC\text{and}△BDC,\angle ABC=\angle BDC={90}^{\circ }\left(given\right)$
$\angle BCA=\angle BCD\text{(Common in both triangles)}$

$\therefore △ABC\sim △BDC\text{(By AA similarity criterion)}$

$\Rightarrow \frac{AC}{BC}=\frac{BC}{DC}\text{(Corresponding sides of similar}\text{triangles are in same ratio)}$

$\Rightarrow AC\times DC=B{C}^2$